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Subspace design [1–3]

Alternative Names: \(q\)-design, Geometric design, Design over finite fields, Design over \(\mathbb{F}_q\), Design in the \(q\)-Johnson scheme, Design in vector spaces.

Description

A constant-dimension code that forms a design on the finite Grassmannian. Subspace designs exist for all parameters in sufficiently large dimension that also satisfies divisibility constraints [4,5].

An alternative but related definition is given in Refs. [6,7].

Notes

See [8] for a review and history of subspace designs.Popular summary of the existence of subspace designs in Quanta Magazine.

Cousin

  • Combinatorial design— Combinatorial designs are designs in Johnson space, the space of all size-\(w\) subsets of a set with \(n\) elements. The \(q\)-Johnson spaces generalize this notion to subspaces and reduce to Johnson spaces at \(q=1\). In other words, combinatorial designs are designs over spaces of subsets, while subspace designs are designs over spaces of subspaces.

Member of code lists

Primary Hierarchy

Classical Domain
Matrix Kingdom
Matrix-based codeECC
Subspace code
Constant-dimension codeECC
Parents
Constant-dimension code
\(t\)-design
Subspace designs are designs on the finite-field Grassmannian (a.k.a. \(q\)-Johnson space or \(q\)-Johnson association scheme) [9][10; Sec. 8.6].
Subspace design

References

  1. [1]

    C. Berge and D. Ray-Chaudhuri, “Unsolved problems”, Lecture Notes in Mathematics 278 (1974) DOI

  2. [2]

    P. J. Cameron, “Generalisation of Fisher’s inequality to fields with more than one element”, Combinatorics, London Mathematical Society Lecture Note Series 13, 9-13 (1973)

  3. [3]

    P. Delsarte, “Association schemes and t-designs in regular semilattices”, Journal of Combinatorial Theory, Series A 20, 230 (1976) DOI

  4. [4]

    A. Fazeli, S. Lovett, and A. Vardy, “Nontrivial t-designs over finite fields exist for all t”, Journal of Combinatorial Theory, Series A 127, 149 (2014) DOI

  5. [5]

    P. Keevash, A. Sah, and M. Sawhney, “The existence of subspace designs”, (2023) arXiv:2212.00870

  6. [6]

    V. Guruswami and S. Kopparty, “Explicit Subspace Designs”, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science 608 (2013) DOI

  7. [7]

    V. Guruswami and C. Xing, “List decoding reed-solomon, algebraic-geometric, and gabidulin subcodes up to the singleton bound”, Proceedings of the forty-fifth annual ACM symposium on Theory of Computing 843 (2013) DOI

  8. [8]

    M. Braun, M. Kiermaier, and A. Wassermann, “q-Analogs of Designs: Subspace Designs”, Signals and Communication Technology 171 (2018) DOI

  9. [9]

    Ph. Delsarte, “Hahn Polynomials, Discrete Harmonics, andt-Designs”, SIAM Journal on Applied Mathematics 34, 157 (1978) DOI

  10. [10]

    T. Ceccherini-Silberstein, F. Scarabotti, and F. Tolli, Harmonic Analysis on Finite Groups (Cambridge University Press, 2008) DOI

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Zoo Code ID: subspace_design

Cite as:
“Subspace design”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/subspace_design, arXiv:2606.11484
BibTeX:
@incollection{eczoo_subspace_design,
title={Subspace design},
booktitle={The Error Correction Zoo},
year={2026},
editor={Albert, Victor V. and Faist, Philippe},
eprint={2606.11484},
doi={10.48550/arXiv.2606.11484},
url={https://errorcorrectionzoo.org/c/subspace_design}
}
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Permanent link:
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Cite as:

“Subspace design”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/subspace_design, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/matrices/subspace/subspace_design.yml.